US11637694B2 - Secret material exchange and authentication cryptography operations - Google Patents
Secret material exchange and authentication cryptography operations Download PDFInfo
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- US11637694B2 US11637694B2 US17/040,949 US201917040949A US11637694B2 US 11637694 B2 US11637694 B2 US 11637694B2 US 201917040949 A US201917040949 A US 201917040949A US 11637694 B2 US11637694 B2 US 11637694B2
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
- H04L9/0816—Key establishment, i.e. cryptographic processes or cryptographic protocols whereby a shared secret becomes available to two or more parties, for subsequent use
- H04L9/0819—Key transport or distribution, i.e. key establishment techniques where one party creates or otherwise obtains a secret value, and securely transfers it to the other(s)
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
- H04L9/0816—Key establishment, i.e. cryptographic processes or cryptographic protocols whereby a shared secret becomes available to two or more parties, for subsequent use
- H04L9/0838—Key agreement, i.e. key establishment technique in which a shared key is derived by parties as a function of information contributed by, or associated with, each of these
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
- H04L9/0816—Key establishment, i.e. cryptographic processes or cryptographic protocols whereby a shared secret becomes available to two or more parties, for subsequent use
- H04L9/085—Secret sharing or secret splitting, e.g. threshold schemes
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/08—Key distribution or management, e.g. generation, sharing or updating, of cryptographic keys or passwords
- H04L9/0861—Generation of secret information including derivation or calculation of cryptographic keys or passwords
- H04L9/0869—Generation of secret information including derivation or calculation of cryptographic keys or passwords involving random numbers or seeds
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/14—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols using a plurality of keys or algorithms
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
- H04L9/30—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy
- H04L9/3093—Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy involving Lattices or polynomial equations, e.g. NTRU scheme
Definitions
- Cryptography is related to the study of protocols, techniques, and approaches that prevent third parties from accessing, reading, and/or interpreting secret data.
- Cryptography can be applied to various processes in information security, such as data integrity and encryption, confidentiality, authentication, verification, and non-repudiation.
- cryptography has several applications in various fields, including data encryption and privacy, computer network communications and transaction processing, and computing system security and integrity.
- Modern cryptography often relies upon computational hardness in mathematical theory. In other words, it might be theoretically possible to break certain cryptographic systems, but the time required to do so makes such cryptographic-defeating processes intractable. Typically, computationally-secure cryptography processes are preferable to those which are easier to defeat. At the same time, however, computationally-secure cryptography processes might be more computationally-intensive to implement and, thus, more time consuming and costly. In that context, although some cryptographic processes, such as a one time pad, cannot be broken or defeated even with unlimited computing power, those schemes are more difficult to implement than a good, theoretically-breakable but computationally secure approach. As such, modern computing devices may exchange secret data using cryptographic processes having security problems (e.g., the processes are susceptible to brute force attack). At the same time, those cryptographic processes may be resource intensive (e.g., the processes are computationally-intensive to implement).
- FIG. 1 illustrates a process of secret text transfer using asymmetric keys.
- FIG. 2 illustrates a representative process of secret key transfer using cryptography processes according to various embodiments described herein.
- FIG. 3 A illustrates an example distribution function of variables resulting from the white noise associative cryptography key operations according to various embodiments described herein.
- FIG. 3 B illustrates example probability distribution functions of variables resulting from the white noise associative cryptography key operations according to various embodiments described herein.
- FIG. 4 illustrates example user interfaces of a program to perform cryptography key operations according to various embodiments described herein.
- FIG. 5 illustrates a more particular example of a secret key transfer process according to the concepts described herein.
- FIG. 6 illustrates an example of a secret key transfer process using authentication according to the concepts described herein.
- cryptography is related to the study of protocols, techniques, and approaches that prevent third parties from accessing, reading, and/or interpreting secret data.
- RSA Rivest-Shamir-Adleman
- ECC elliptic curve cryptography
- Those cryptosystems are based on complexity and can, theoretically, be decrypted.
- the cryptographic processes described herein is more immune to cryptanalysis and permits the sharing of secret data, such as symmetric keys and other secret data, over public networks.
- the cryptographic system can also be used for authentication. No known methods of traditional or quantum computing can be used to circumvent the cryptographic approaches described herein.
- the cryptographic system described herein was developed to achieve a number of goals including (1) securely exchanging cryptographic keys over public networks, (2) information ciphering, authentication, and (4) encryption for public networks that is secure against standard and quantum computing.
- white noise can be defined as (or can include) a sequence of independent random variables (e.g., discrete numbers) with a uniform probability distribution.
- Polynomial white noise can be defined as (or can include) a sequence of polynomial function values composed by independent random variables (e.g., discrete numbers) with a uniform probability distribution.
- No known algorithm can decrypt the operations described herein due, at least in part, to the use of white noise randomization.
- the unknown independent variables appear to third parties as random white noise and, thus, there is no correlation between those variables and any information being transferred.
- the key exchange method or process described herein can be shown as an exchange of matrices with a corresponding number of different unknown independent variables and visible values.
- the number of unknown independent variables always exceeds the number of visible independent values in any combination of subsets of matrices.
- the number of unknown variables exceeds the number of publically visible polynomial functions.
- no inverse polynomial functions can be determined without information about the secret key—even if the plain text of the secret key is known by a third party.
- FIG. 1 illustrates a process of secret text transfer using asymmetric keys.
- Alice wishes to communicate secret text to Bob over a public network, such as the Internet
- Eve is the eavesdropper.
- the secret text which can be a symmetric key or any other secret information
- Alice and Bob use asymmetric cryptography.
- Asymmetric cryptography relies upon a key pair including a public key that can be disseminated to third parties (e.g., Alice) and a private key which is kept private (e.g., by Bob).
- any person can encrypt a message using the public key, and that encrypted message can only be decrypted using the private key.
- the strength of asymmetric cryptography relies on the degree of difficulty (e.g., computational impracticality) for a private key to be determined from its associated public key.
- Asymmetric cryptography also depends on keeping the private key private.
- Alice obtains a copy of a public key from Bob (or any other source).
- Alice encrypts the secret text using the public key to produce the encrypted secret text and communicates it to Bob over the public network.
- Bob then decrypts the encrypted secret text using the private key to obtain the secret key.
- Eve can only see the encrypted secret text. Even if Eve obtains a copy of the encrypted secret text and the public key used to create it, Eve cannot obtain the secret text from the encrypted secret text using the public key. Instead, only the private key, which is securely held and protected by Bob, can be used to decrypt the encrypted secret text to obtain the secret text from Alice.
- asymmetric cryptography there are drawbacks and limitations to using asymmetric cryptography. For example, it is algorithmically possible to estimate (or determine) the private key in a key pair from the publicly available public key. Additionally, asymmetric key pairs are relatively difficult and time consuming to create, typically depending upon the identification of large prime numbers. Further, asymmetric cryptography can be vulnerable in that it may produce the same predictable encrypted output when the same secret text is encrypted.
- a first cryptographic function is applied to secret data.
- the first cryptographic function operates as a type of cryptographic key and encrypts or ciphers the secret data to produce a first encrypted result.
- the first encrypted result can be securely transmitted by a first device to a second device.
- the second device applies a second cryptographic function to the first encrypted result.
- the second cryptographic function operates as a cryptographic key and further (or doubly) encrypts or ciphers the first encrypted result to produce a second (or doubly) encrypted result.
- the secret data has been encrypted by two different cryptographic functions, each of them being sufficient to secure the secret data.
- the two different cryptographic functions can then be inversed or removed, in any order, to reveal the secret data.
- FIG. 2 illustrates a representative process 20 of secret key transfer using cryptography processes according to various embodiments described herein.
- the process described below can be performed by any suitable computing device(s) including a processor and memory, without limitation.
- Alice wants to securely pass the secret key X to Bob over a public network. To do so, Alice should first encrypt the secret key X before sending it to Bob.
- the cryptographic function F A can be embodied as any suitable mathematical function having an inverse which cannot be determined without knowledge of a certain set of parameters of the mathematical function.
- the function F A can be embodied as a polynomial function or multivariate polynomial function defined in part by one or more variables, combinations of variables, combinations of variables at various powers, and coefficients.
- Alice also holds a first inverse cryptographic function F ⁇ 1 A.
- the process 20 includes Alice generating, with a first computing device, a first random lock X A .
- the first random lock X A can be embodied as an array or vector of random scalar integers, for example, or another suitable organized structure of random numbers.
- the first random lock X A can operate as a type of initialization vector upon which the cryptographic function F A is applied in combination with the secret key X.
- the first random lock X A helps to randomize the application of the cryptographic function F A creating, in effect, a new random cryptographic function F A for each different random lock X A .
- the first random lock X 4 helps to achieve semantic security, so that repeated usage of the cryptographic function F A with the same operand does not produce the same ciphered result and does not allow an attacker to infer any information.
- the process 20 includes Alice applying, with the first computing device, the first cryptographic function F A to a combination of the secret key X and the first random lock X A to produce a first encrypted result R 1 .
- Alice's secret key X which can include letters, numbers, American Standard Code for Information Interchange (ASCII) characters, etc.
- ASCII American Standard Code for Information Interchange
- the cryptographic function F A can be embodied as any suitable mathematical function, such as a polynomial or multivariate polynomial function.
- the cryptographic function F A can be embodied as a polynomial function F(CX k ) of kth order written as:
- Alice's secret key X which may include letters, numbers, American Standard Code for Information Interchange (ASCII) characters, etc.
- ASCII American Standard Code for Information Interchange
- FIG. 3 A a distribution function of the variables in the results R 1 , R 2 , and R 3 is shown in FIG. 3 A
- probability distribution functions of the variables in the results R 1 , R 2 , and R 3 is shown in FIG. 3 B .
- the structure of the polynomial function F(CX k ) and the coefficients can be known to others (although they generally are not) from the formalization of the algorithm. However, even if the structure of the polynomial function F and values of the coefficients C, k are known to a third party, the third party still cannot decrypt the transferred information.
- the process 20 includes Alice transmitting, with the first computing device, the first encrypted result R 1 to Bob's second computing device.
- the process 20 includes Bob generating, with the second computing device, a second random lock X B .
- the second random lock X B can be embodied as an array or vector of random scalar integers, for example, or another suitable organized structure of random numbers.
- the second random lock Xs can also operate as a type of initialization vector for the cryptographic function F B .
- the second random lock X B helps to randomize the application of Bob's cryptographic function F B creating, in effect, a new random cryptographic function F B for each different random lock XB .
- the second random lock X B helps to achieve semantic security, so that repeated usage of the cryptographic function F B with the same operand does not produce the same ciphered result and does not allow an attacker to infer any information.
- the process includes Bob applying, with the second computing device, Bob's cryptographic function F B to a combination of the first encrypted result R 1 and the second random lock X B to produce a second encrypted result R 2 .
- the first encrypted result R 1 e.g., F A (X,X A )
- the cryptographic function F B can be embodied as any suitable mathematical function, such as a polynomial or multivariate polynomial function.
- the cryptographic function F B can be embodied as a polynomial function F(CX k ) of kth order according to that shown above in Equation (1).
- Alice's secret key X has been encrypted or ciphered by two different cryptographic functions F A and F B , each of them being sufficient to secure the secret key X from others.
- the two different cryptographic functions can then be inversed or removed, in any order, to reveal the secret key X.
- decrypt the secret key X from the second encrypted result R 2 i.e., to undo the effects of the cryptographic functions F A and Fa
- the order in which the second encrypted result R 2 is applied to the inverse cryptographic functions F ⁇ 1 A and F ⁇ 1 B . does not impact the results of the decryption of secret key X from the second encrypted result R 2 .
- any number of cryptographic functions to F 1 . . . F N can be applied to encrypt secret data in any order to produce an encrypted result R N , and that encrypted result R N can be decrypted in any order using the inverse cryptographic functions F ⁇ 1 1 . . . F ⁇ 1 N .
- the process 20 includes Bob transmitting, with the second computing device, the second encrypted result R 2 to the first computing device.
- the process 20 includes Alice applying, with the first computing device, the first inverse cryptographic function F ⁇ 1 A to the second encrypted result R 2 to produce the result R 3 .
- the first inverse cryptographic function F ⁇ 1 A unlocks or removes the effect of both the first random lock X A and the first cryptographic function F A .
- the result R 3 is what remains of the second encrypted result R 2 after the effect of the first random lock X A and the first cryptographic function F A are undone or unlocked (e.g., F B (X,X B )).
- the result R 3 is still encrypted, but only by Bob's second random lock X B and the second cryptographic function F B , and the result R 3 can be securely transmitted over the public network.
- the process 20 includes Alice transmitting, with the first computing device, the result R 3 to the second computing device.
- the process 20 includes Bob applying, with the second computing device, the second inverse cryptographic function F ⁇ 1 B to the result R 3 to arrive at the secret key X.
- the secret key X has been securely communicated from Alice to Bob.
- key pairs are not used in the process 20 .
- the method is cryptanalysis resistant. To obtain the only solution x 1 , .
- the third party e.g., outsider Eve
- plain text (as a letter or ASCII code of 256 numbers) is represented in ciphered text by three corresponding random numbers r 1 , r 2 and r 3 which are calculated by a random generator.
- Table 2 shows an example of how the plain text “This is a plain text” appears in ciphered numbers.
- Ciphered text text r 1 r 2 r 3 T 0.001251 0.563585 0.003585 h 0.193304 0.808741 0.158307 i 0.585009 0.479873 0.28051 s 0.350291 0.895962 0.313555 0.82284 0.746605 0.614412 i 0.174108 0.858943 0.151801 s 0.710501 0.513535 0.363394 0.303995 0.014985 0.006167 a 0.091403 0.364452 0.035009 0.147313 0.165899 0.02575 p 0.988525 0.445692 0.438709 1 0.119083 0.004669 0.001204 i a 0.00891 1 0.37788 0.005292 i 0.531663 0.571184 0.303183 n 0.601764 0.607166 0.363988 0.166234 0.663045 0.113037 t 0.450789 0.352123 0.159469 e 0.057039 0.
- Uniform distribution is called “white noise” due to its informative features.
- the correlation function between any two variables x and y is estimated as follows:
- corr ( x , y ) ⁇ ⁇ ( x - x _ ) ⁇ ( y - y _ ) ⁇ ⁇ ( x - x _ ) 2 ⁇ ⁇ ⁇ ( y - y _ ) 2 .
- the results of correlation function evaluation for pairs (r 1 , r 2 ), (r 2 , r 3 ) and (r 1 , r 3 ) are given in Table 3 below.
- the correlation is negligibly small, which means that ciphered information is encapsulated into white noise and is not analyzable by a third party. There are no known algorithms to decrypt the ciphered information without the encryption key.
- a computer program was developed to implement the method described herein. As shown in FIG. 4 , Alice securely sends her secret text “Hello bob” to Bob using the three pass transaction. In FIG. 4 , random values appear to a third party during the three pass transaction (specially shown in the blue box).
- the processes described herein can be used to achieve unbreakable (or nearly unbreakable) encryption over wireless, wired, and public networks, and against quantum computing attacks. It requires relatively little processing power for encrypting and decrypting and, thus, can be used for rapid verification and transactions. A practically limitless number of new keys can be generated on the fly. Thus, the keys can be changed on every transaction. Encryption and decryption can also occur on individual devices due to the high speed of encryption and low processing requirements. Further, there is no single point of compromise because every individual party has their own key. If a key is compromised, it is the one compromised and can be renewed or replaced.
- FIG. 5 illustrates a more particular example of a secret key transfer process 30 according to the concepts described herein. While an example using square matrices of a certain size is provided below, the concepts described herein can be extended to use with square matrices of any size. Further, although the example below is presented in certain cases as steps between “Alice” and “Bob,” the process is conducted by computing systems or devices.
- X ⁇ x 1 x 2 x 3 x 4 ⁇ ⁇ ... ⁇ ⁇ x m - 3 x m - 2 x m - 1 x m ⁇
- the last matrix is not fully filled in. In this case, the rest of the matrix members can be generated and added as any random numbers without influencing the algorithm.
- X Z 1 ⁇ Z 2
- ⁇ Z 1 ⁇ Z 1 Z 2 Z 3 Z 2 ⁇ Z 3 Z 1 ⁇
- Z 2 ⁇ Z 4 Z 5 Z 6 Z 4 ⁇ Z 5 Z 6 ⁇
- the inverse of matrix X, or X ⁇ 1 does not exist (see properties of singular matrices and matrix determinants in APPENDIX).
- the matrix X represents a portion of the secret key K, ⁇ k 1 , k 2 , k 3 ⁇ .
- the process further includes generating a uniformly distributed random matrices Y 1 , Y 2 and inverse matrices Y 1 ⁇ 1 , Y 2 ⁇ 1 , as follows:
- Y 1 ⁇ y 1 y 2 y 3 y 4 ⁇
- ⁇ Y 1 - 1 ⁇ y 4 - y 2 - y 3 y 1 ⁇ y 1 ⁇ y 4 - y 2 ⁇ y 3 , y i ⁇ R , y 1 ⁇ y 4 ⁇ y 2 ⁇ y 3
- ⁇ Y 2 ⁇ y 5 y 6 y 7 y 8 ⁇
- Y 2 - 1 ⁇ y 8 - y 6 - y 7 y 5 ⁇ y 5 ⁇ y 8 - y 6 ⁇ y 7 , y i ⁇ R , y 5 ⁇ y 8 ⁇ y 6 ⁇ y 7 .
- the process also includes generating uniformly distributed random centrosymmetric A 1 , A 2 , B 1 , B 2 , and inverse A 1 ⁇ 1 , A 2 ⁇ 1 , B 1 ⁇ 1 , B 2 ⁇ 1 matrices as follows:
- the process includes Alice generating and sending matrices M 1 and M 2 to Bob, as follows:
- M 1 ⁇ m 1 ( 1 ) m 2 ( 1 ) m 3 ( 1 ) m 4 ( 1 ) ⁇
- ⁇ M 2 ⁇ m 1 ( 2 ) m 2 ( 2 ) m 3 ( 2 ) m 4 ( 2 ) ⁇
- ⁇ M 3 ⁇ m 1 ( 3 ) m 2 ( 3 ) m 3 ( 3 ) m 4 ( 3 ) ⁇
- Alice sends to Bob fourteen publicly visible values (m 1 (1) , m 2 (1) , m 3 (1) , m 4 (1) , m 1 (2) , m 2 (2) , m 3 (2) , m 1 (3) , m 2 (3) , m 3 (3) , m 4 (3) , m 1 (4) , m 2 (4) , m 3 (4) ) of matrices M 1 , M 2 , M 3 , and M 4 that are calculated from twenty-two independent unknown (for the third party) variables (a 1 , a 2 , a 3 , a 4 , b 1 , b 2 , b 3 , b 4 , y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , z 1 , z 2 , z 3 , z 4 , z 5 , z 6 ) known by Alice only,
- ⁇ m 1 ( 1 ) a 1 ⁇ y 1 + a 2 ⁇ y 2
- ⁇ m 2 ( 1 ) a 2 ⁇ y 1 + a 1 ⁇ y 2
- ⁇ m 3 ( 1 ) a 1 ⁇ y 3 + a 2 ⁇ y 4
- ⁇ m 4 ( 1 ) a 2 ⁇ y 3 + a 1 ⁇ y 4
- ⁇ m 1 ( 2 ) b 1 ⁇ ( x 1 ⁇ y 4 - x 3 ⁇ y 2 ) + b 2 ⁇ ( x 3 ⁇ y 1 - x 1 ⁇ y 3 ) y 1 ⁇ y 4 - y 2 ⁇ y 3
- ⁇ m 2 ( 2 ) b 1 ⁇ ( x 2 ⁇ y 4 - y 2 ⁇ x 2 ⁇ / ⁇ x 1 ) + b 2 ⁇ (
- the process includes Bob receiving the M 1 and M 2 matrices from Alice.
- the process includes generating uniformly distributed random centrosymmetric matrices C 1 , C 2 and inverse C 1 ⁇ 1 , C 2 ⁇ 1 matrices, as follows:
- C 1 ⁇ c 1 c 2 c 2 c 1 ⁇
- ⁇ C 2 ⁇ c 3 c 4 c 4 c 3 ⁇
- ⁇ C 1 - 1 ⁇ c 1 - c 2 - c 2 c 1 ⁇ c 1 2 - c 2 2 , c i ⁇ R , c 1 2 ⁇ c 2 2
- C 2 - 1 ⁇ c 3 - c 4 - c 4 c 3 ⁇ c 3 2 - c 4 2 , c i ⁇ R , c 3 2 ⁇ c 4 2 .
- the process at step 306 also includes generating uniformly distributed random matrices D and H, as follows:
- D ⁇ d 1 d d 3 d 4 ⁇
- H ⁇ h 1 h 2 h 3 h 4 ⁇ , d i ⁇ h i ⁇ R , d 1 ⁇ d 4 ⁇ d 2 ⁇ d 3 , h 1 ⁇ h 4 ⁇ h 2 ⁇ h 3 .
- the process at step 306 also includes generating corresponding inverse matrices D ⁇ 1 and H ⁇ 1 , as follows:
- D - 1 ⁇ d 1 d 2 d 3 d 4 ⁇ d 1 ⁇ d 4 - d 2 ⁇ d 3
- H - 1 ⁇ h 1 h 2 h 3 h 4 ⁇ h 1 ⁇ h 4 - h 2 ⁇ h 3 .
- the process at step 306 also includes generating the matrices M 5 , M 6 , M 7 and M 8 , as follows:
- M 5 ⁇ m 1 ( 5 ) m 2 ( 5 ) m 3 ( 5 ) m 4 ( 5 ) ⁇
- ⁇ M 6 ⁇ m 1 ( 6 ) m 2 ( 6 ) m 3 ( 6 ) m 4 ( 6 ) ⁇
- ⁇ M 7 ⁇ m 1 ( 7 ) m 2 ( 7 ) m 3 ( 7 ) m 4 ( 7 ) ⁇
- M 8 ⁇ m 1 ( 8 ) m 2 ( 8 ) m 3 ( 8 ) m 4 ( 8 ) ⁇ as a result of the following calculations:
- the process includes Bob sending to Alice fourteen publicly visible values (m 1 (5) , m 2 (5) , m 3 (5) , m 1 (6) , m 2 (6) , m 3 (6) , m 1 (7) , m 2 (7) , m 3 (7) , m 4 (7) , m 1 (8) , m 2 (8) , m 3 (8) ) of matrices M 3 and M 4 that are calculated from sixteen independent unknown (for the third party) variables (c 1 , c 2 , c 3 , c 4 , d 1 , d 2 , d 3 , d 4 , e 1 , e 2 , e 3 , e 4 , h 1 , h 2 , h 3 , h 4 ) which are known by Bob only, as follows:
- the process includes multiplying the results of those together to arrive at the matrix M 5 , as follows:
- the entire scheme of the key exchange process can be performed using an exchange of matrices with a corresponding number of different unknown independent variables (underlined in Table 6) and visible (by the third party) values (bolded in Table 6).
- This scheme demonstrates that the number of unknown independent variables always exceeds the number of visible independent values in any combination of subsets of matrices.
- M 2 B 1 Y 1 ⁇ 1 Z 1
- M 4 B 2 Y 2 ⁇ 1 Z 2
- M 6 C 1 B 1 Y 1 ⁇ 1 Z 1 E
- M 6 C 2 B 2 Y 2 ⁇ 1 Z 2 H
- M 9 DZ 1 Z 2 H
- FIG. 6 illustrates an example secret material or key exchanging process using authentication according to the concepts described herein.
- the process 40 includes Alice generating uniformly distributed random matrices Y 1 , Y 2 and inverse matrices Y 1 ⁇ 1 , Y 2 ⁇ 1 , as follows:
- Y 1 ⁇ " ⁇ [LeftBracketingBar]” y 1 y 2 y 3 y 4 ⁇ " ⁇ [RightBracketingBar]”
- Y 1 - 1 ⁇ " ⁇ [LeftBracketingBar]” y 1 - y 2 - y 3 y 4 ⁇ " ⁇ [RightBracketingBar]” y 1 ⁇ y 4 - y 2 ⁇ y 3 , y i ⁇ R , y 1 ⁇ y 4 ⁇ y 2 ⁇ y 3
- Y 2 ⁇ " ⁇ [LeftBracketingBar]” y 5 y 6 y 7 y 8 ⁇ " ⁇ [RightBracketingBar]”
- ⁇ Y 2 - 1 ⁇ " ⁇ [LeftBracketingBar]” y 8 - y 6 - y 7 y 5 ⁇ " ⁇ [RightBracketingBar]” y 5 ⁇ y 8 - y 6
- Alice also generates uniformly distributed random centrosymmetric matrices A and B, as follows:
- a 1 ⁇ " ⁇ [LeftBracketingBar]” a 1 a 2 a 2 a 1 ⁇ " ⁇ [RightBracketingBar]”
- a 2 ⁇ " ⁇ [LeftBracketingBar]” a 3 a 4 a 4 a 3 ⁇ " ⁇ [RightBracketingBar]”
- a 1 - 1 ⁇ " ⁇ [LeftBracketingBar]” a 1 - a 2 - a 2 a 1 ⁇ " ⁇ [RightBracketingBar]” a 1 2 - a 2 2
- the process includes Alice sending to Bob results as matrices M 1 and M 2 , as follows:
- M 1 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 1 ) m 2 ( 1 ) m 3 ( 1 ) m 4 ( 1 ) ⁇ " ⁇ [RightBracketingBar]”
- M 2 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 2 ) m 2 ( 2 ) m 3 ( 2 ) m 4 ( 2 ) ⁇ " ⁇ [RightBracketingBar]”
- M 3 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 3 ) m 2 ( 3 ) m 3 ( 3 ) m 4 ( 3 ) ⁇ " ⁇ [RightBracketingBar]”
- ⁇ M 4 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 4 ) m 2 ( 4 ) m 3 ( 4 ) ⁇ " ⁇ [RightBracketingBar]”
- ⁇ M 4 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 4
- Bob receives M 1 and M 2 from Alice.
- Bob generates uniformly distributed random centrosymmetric matrices C 1 , C 2 and inverse C 1 ⁇ 1 , C 2 ⁇ 1 matrices, as follows:
- D ⁇ " ⁇ [LeftBracketingBar]” d 1 d 2 d 3 d 4 ⁇ " ⁇ [RightBracketingBar]”
- ⁇ H ⁇ " ⁇ [LeftBracketingBar]” h 1 h 2 h 3 h 4 ⁇ " ⁇ [RightBracketingBar]” , d i , h i ⁇ R , d 1 ⁇ d 4 ⁇ d 2 ⁇ d 3 , h 1 ⁇ h 4 ⁇ h 2 ⁇ h 3 , and correspondent inverse matrices D ⁇ 1 and H ⁇ 1 , as follows:
- Bob also obtains the matrices M 5 , M 6 , M 7 and M 8 , defined as follows:
- M 5 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 5 ) m 2 ( 5 ) m 3 ( 5 ) m 4 ( 5 ) ⁇ " ⁇ [RightBracketingBar]”
- M 6 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 6 ) m 2 ( 6 ) m 3 ( 6 ) m 4 ( 6 ) ⁇ " ⁇ [RightBracketingBar]”
- M 7 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 7 ) m 2 ( 7 ) m 3 ( 7 ) m 4 ( 7 ) ⁇ " ⁇ [RightBracketingBar]”
- the process includes Alice generating a uniformly distributed random matrix G, as follows:
- G ⁇ " ⁇ [LeftBracketingBar]” g 1 g 2 g 3 g 4 ⁇ " ⁇ [RightBracketingBar]” , g i ⁇ R .
- the process also includes Alice multiplying both the matrices M 5 , M 6 , M 7 and M 8 with the inverse matrices which are known to her, A 1 ⁇ 1 , A 2 ⁇ 1 , B 1 ⁇ 1 and B 2 ⁇ 1 , respectively, as follows:
- the process includes Alice sending three publicly visible values to Bob, including (m 1 (9) , m 2 (9) , m 3 (9) ).
- Alice also sends four publicly visible values to Ed (m 1 (6) , m 2 (6) , m 3 (6) , m 4 (6) ) of the matrix M 10 , defined as:
- M 10 ⁇ " ⁇ [LeftBracketingBar]” m 1 ( 10 ) m 2 ( 10 ) m 3 ( 10 ) m 4 ( 10 ) ⁇ " ⁇ [RightBracketingBar]” , as a result of the following calculations: M 10 ⁇ N A G. (9B)
- Ed receives the matrix M6 from Alice.
- Bob also receives the matrix M 9 from Alice at step 410 .
- the embodiments described herein can be implemented by either a method or process or as a system or device.
- the method can be performed using any suitable computing device, and the system can be embodied as any suitable computing device.
- the computing device can include at least one processing system, for example, having one or more processors and memories electrically and communicatively coupled together using a local interface.
- the local interface can be embodied as a data bus with an accompanying address/control bus or other addressing, control, and/or command lines.
- the memory can store data and software or executable code components executable by the processor.
- the memory can store executable-code components associated with cryptographic operations for execution by the processor.
- the software or executable-code components can be developed using or embodied in various programming languages, such as, for example, C, C++, C#, Objective C, JAVA®, JAVASCRIPT®, Perl, PHP, VISUAL BASIC®, PYTHON®, RUBY, FLASH®, or other programming languages.
- executable or “for execution” refer to software forms that can ultimately be run or executed by a processor, whether in source, object, machine, or other form.
- executable programs include, for example, a compiled program that can be translated into a machine code format and loaded into a random access portion of memory and executed by a processor, source code that can be expressed in an object code format and loaded into a random access portion of the memory and executed by the processor, or source code that can be interpreted by another executable program to generate instructions in a random access portion of the memory and executed by the processor, etc.
- An executable program can be stored in any portion or component of the memory including, for example, a random access memory (RAM), read-only memory (ROM), magnetic or other hard disk drive, solid-state, semiconductor, or similar drive, universal serial bus (USB) flash drive, memory card, optical disc (e.g., compact disc (CD)) or digital versatile disc (DVD)), floppy disk, magnetic tape, or other memory component.
- RAM random access memory
- ROM read-only memory
- magnetic or other hard disk drive solid-state, semiconductor, or similar drive
- USB universal serial bus
- memory card e.g., compact disc (CD)) or digital versatile disc (DVD)
- CD compact disc
- DVD digital versatile disc
- FIGS. 2 and 5 illustrate a certain order, it is understood that the order can differ from that which is depicted. For example, an order of execution of two or more blocks can be scrambled relative to the order shown. Also, two or more blocks shown in succession can be executed concurrently or with partial concurrence. Further, in some embodiments, one or more of the blocks can be skipped or omitted. In addition, any number of counters, state variables, warning semaphores, or messages might be added to the logical flow described herein, for purposes of enhanced utility, accounting, performance measurement, or providing troubleshooting aids, etc. It is understood that all such variations are within the scope of the present disclosure.
- any algorithm, method, process, or logic described herein that are embodied, at least in part, by software or executable-code components can be embodied or stored in any tangible or non-transitory computer-readable medium or device for execution by an instruction execution system such as a general purpose processor.
- the logic can be embodied as, for example, software or executable-code components that can be fetched from the computer-readable medium and executed by the instruction execution system.
- the instruction execution system can be directed by execution of the instructions to perform certain processes such as those illustrated in FIG. 2 .
- a “computer-readable medium” can be any tangible medium that can contain, store, or maintain any logic, application, software, or executable-code component described herein for use by or in connection with an instruction execution system.
- the computer-readable medium can include any physical media such as, for example, magnetic, optical, or semiconductor media. More specific examples of suitable computer-readable media include, but are not limited to, magnetic tapes, magnetic floppy diskettes, magnetic hard drives, memory cards, solid-state drives, USB flash drives, or optical discs. Also, the computer-readable medium can include a RAM including, for example, an SRAM, DRAM, or MRAM. In addition, the computer-readable medium can include a ROM, a PROM, an EPROM, an EEPROM, or other similar memory device.
- Disjunctive language such as the phrase “at least one of X, Y, or Z,” unless specifically stated otherwise, is to be understood with the context as used in general to present that an item, term, etc., can be either X, Y, or Z, or any combination thereof (e.g., X, Y, and/or Z). Thus, such disjunctive language is not generally intended to, and should not, imply that certain embodiments require at least one of X, at least one of Y, or at least one of Z to be each present.
- centrosymmetric matrix A is a matrix which is symmetric about its center.
- a centrosymmetric matrix A has the following form:
- a square matrix is singular if and only if its determinant is 0. Because a square matrix formed from a random distribution of values will almost never be singular, singular matrices are rare.
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Abstract
Description
where Ci . . . k are coefficients of the polynomial function F(CXk), and Xi . . . k are combinations of the operand X, which can include a combination of a random lock and secret data.
TABLE 1 | ||
Public-Private Key | ||
Asymmetrical | PWN Three | |
Features | (RSA, ECC) | Pass Method |
Numbers | Prime Numbers | Any Random |
Numbers | ||
Time to Develop New | Relatively More Costly | Negligible |
Key | ||
Processing Time | Relatively More Costly | Negligible |
Inverse Function From | Relatively Complex | Inverse Function |
Public Key | Does Not Exist | |
Third Party Defeat | Possible | Never |
Public Network Output | Constant, predictable | Random, |
For Constant Input | unpredictable | |
TABLE 2 | |||
Plain | |||
text | Ciphered text |
text | r1 | r2 | r3 | ||
T | 0.001251 | 0.563585 | 0.003585 | ||
h | 0.193304 | 0.808741 | 0.158307 | ||
i | 0.585009 | 0.479873 | 0.28051 | ||
s | 0.350291 | 0.895962 | 0.313555 | ||
0.82284 | 0.746605 | 0.614412 | |||
i | 0.174108 | 0.858943 | 0.151801 | ||
s | 0.710501 | 0.513535 | 0.363394 | ||
0.303995 | 0.014985 | 0.006167 | |||
a | 0.091403 | 0.364452 | 0.035009 | ||
0.147313 | 0.165899 | 0.02575 | |||
p | 0.988525 | 0.445692 | 0.438709 | ||
1 | 0.119083 | 0.004669 | 0.001204 i | ||
a | 0.00891 1 | 0.37788 | 0.005292 | ||
i | 0.531663 | 0.571184 | 0.303183 | ||
n | 0.601764 | 0.607166 | 0.363988 | ||
0.166234 | 0.663045 | 0.113037 | |||
t | 0.450789 | 0.352123 | 0.159469 | ||
e | 0.057039 | 0.607685 | 0.037377 | ||
x | 0.783319 | 0.802606 | 0.623152 | ||
t | 0.519883 | 0.30195 | 0.157851 | ||
The results of correlation function evaluation for pairs (r1, r2), (r2, r3) and (r1, r3) are given in Table 3 below.
TABLE 3 | ||||
corr (r1, r2) | corr (r2, r3) | corr (r1, r3) | ||
−0.013927 | −0.002873 | −0.010771 | ||
TABLE 4 | ||
WNT One Pass Transaction | ||
Symmetrical | (in combination with Three | |
Features | FIPS Pub 197 | Pass Transaction) |
Encryption Key Rotation | Must Have | Not Needed |
Processing time | Costly | Negligible |
Security resistance and key | Strong relation | No Relation |
length | ||
Hack | Costly | Never (Potentially |
Impossible) | ||
Public net output for | Constant, | Random, Unpredictable |
constant input (without key | Predictable | |
rotation) | ||
TABLE 5 | |||
Problem | Solution | ||
Establishing a secure and | Digital ID system in the cloud | ||
reliable ID for all transactions | for processing Ids | ||
ID system only used for | |||
registration and verification | |||
Information unhackable | |||
Having a secure payment | Payment system using ID | ||
system that eliminates fraud | Email, internet banking, wireless | ||
transaction | |||
Cryptocurrency that is secure | Absolutely secure, stable, and | ||
and stable | based on verifiable IDs | ||
Fast enough and secure trading | Rapid trading and verification | ||
system for cryptocurrencies | Trading exchanges connected to | ||
Exchange | |||
Mobile Payments | Integrity over wireless signals | ||
and public net | |||
Transactions cannot be defrauded | |||
via screening or copying | |||
Key Management System | Cloud key management service | ||
ID system to outsource all key | |||
management responsibilities | |||
People forget passwords and | Pass eliminates the use of | ||
passwords are a weak point | passwords using ID center | ||
in security | |||
K={k 1 ,k 2 , . . . ,k m},0≤k i≤255.
X=Φ(K),Φ:N m →R m and
X={x 1 ,x 2 , . . . ,x m },x i ϵR.
The matrix X decomposes into two singular matrices Z1 and Z2
Centrosymmetric square matrices A and B are always of the form AB=BA.
which are generated according to the following calculations:
M 1 =Y 1 A 1, (3)
M 2 =B 1 Y 1 −1 Z 1, (4)
M 3 =Y 2 A 2, and (5)
M 4 =B 2 Y 2 −1 Z 3. (6)
as a result of the following calculations:
M 5 =DM 1 C 1 −1 =D 1 Y 1 A 1 C 1 −1, (7)
M 6 =C 1 M 2 E=C 1 B 1 Y 1 −1 Z 1 E, (8)
M 7 =E −1 M 3 C 2 −1 =E −1 YA 2 C 2 −1, and (9)
M 8 =C 2 M 4 H=C 2 B 2 Y 2 −1 Z 2 H. (10)
M 5 =DY 1 A 1 C 1 −1,
M 6 =C 1 B 1 Y 1 −1 Z 1 E,
M 7 =E −1 Y 2 A 2 C 2 −1, and
M 8 =CBY −1 XH.
AC −1 =C −1 A and
CB=BC,
meaning that the matrices M5, M6, M7, and M8 can be transformed into:
M 5 =DY 1 A 1 C 1 −1 =DY 1 C 1 −1 A 1,
M 6 =C 1 B 1 Y 1 −1 Z 1 E=B 1 C 1 Y 1 −1 Z 1 E,
M 7 =E −1 Y 2 A 2 C 2 −1 =E −1 Y 2 C 2 −1 A 2, and
M 8 =C 2 B 2 Y 2 −1 Z 2 H=B 2 C 2 Y 2 −1 Z 2 H,
M 5 A 1 −1 =DY 1 C 1 −1 A 1 A 1 −1 =DY 1 C 1 −1,
B 1 −1 M 6 =B 1 −1 B 1 C 1 Y 1 −1 Z 1 E=C 1 Y 1 −1 Z 1 E,
M 7 A 2 −1 =E −1 Y 2 C 2 −1 A 2 A 2 −1 =E −1 Y 2 C 2 −1, and
B 2 −1 M 8 =B 2 −1 B 2 C 2 Y 2 −1 Z 2 H=C 2 Y 2 −1 Z 2 H.
m 1 (9)=(d 1 x 1 +d 2 x 3)h 1+(d 1 x 2 +d 2 x 4)h 3,
m 2 (9)=(d 1 x 1 +d 2 x 3)h 2+(d 1 x 2 +d 2 x 4)h 4,
m 3 (9)=(d 3 x 1 +d 4 x 3)h 1+(d 3 x 2 +d 4 x 4)h 3, and
m 4 (9) =m 3 (9) m 2 (9) /m 1 (9).
M 9 =DXH.
D −1 M 9 H −1 =D −1 DXHH −1 =X.
TABLE 6 | ||||
Independent | ||||
Variables | Variables | Values | ||
1 | Alice | Y1A1 | A1[2], Y1[4] | 22 | M1[4] | 4 | 14 |
B1Y1 −1Z1 | B1[2], Z1[3] | M2[4] | 3 | ||||
Y2A2 | A2[2], Y2[4] | M3[4] | 4 | ||||
B2Y2 −1Z2 | B2[2], Z2[3] | M4[4] | 3 | ||||
2 | Bob | DY1A1C1 −1 | D [4], C1[2] | 16 | M5[4] | 4 | 14 |
C1B1Y1 −1 Z1E | E [4] | M6[4] | 3 | ||||
E−1Y2A2C2 −1 | M7[4] | 4 | |||||
C2B2Y2 −1 Z2H | C2[2], H [4] | M8[4] | 3 | ||||
3 | Alice | DXH | M9[3] | 3 | 3 |
Total | 38 | 31 | |||
M 2 =B 1 Y 1 −1 Z 1,
M 4 =B 2 Y 2 −1 Z 2,
M 6 =C 1 B 1 Y 1 −1 Z 1 E,
M 6 =C 2 B 2 Y 2 −1 Z 2 H, and
M 9 =DZ 1 Z 2 H
are also singular (due to the matrices Z1 and Z2 being singular).
is used to represent key K={k1, k2, k3}, where x4=x2x3/x1.
of the following calculations:
M 1 =Y 1 A 1, (1B)
M 2 =B 1 Y 1 −1 Z 1, (2B)
M 3 =Y 2 A 2, (3B)
M 4 =B 2 Y 2 −1 Z 2. (4B)
and uniformly distributed random matrices D and H, as follows:
and correspondent inverse matrices D−1 and H−1, as follows:
as a result of the following calculations:
M 5 =DM 1 C 1 −1 =D 1 Y 1 A 1 C 1 −1, (5B)
M 6 =C 1 M 2 E=C 1 B 1 Y 1 −1 Z 1 E, (6B)
M 7 =E −1 M 3 C 2 −1 =E −1 YA 2 C 2 −1, and (7B)
M 8 =C 2 M 4 H=C 2 B 2 Y 2 −1 Z 2 H, (8B)
M 5 =DY 1 A 1 C 1 −1 =DY 1 C 1 −1 A 1,
M 6 =C 1 B 1 Y 1 −1 Z 1 E=B 1 C 1 Y 1 −1 Z 1 E,
M 7 =E −1 Y 2 A 2 C 2 −1 =E −1 Y 2 C 2 −1 A 2, and
M 8 =C 2 B 2 Y 2 −1 Z 2 H=B 2 C 2 Y 2 −1 Z 2 H,
as a result of the following calculations:
M 10 −N A G. (9B)
M 11 =N B(N A)−1 N A G=N B G,
M 11 =N B G. (10B).
G=(N B)−1 M 11=(N B)−1 N B G.
D −1 G −1 M 9 H −1 =D −1 G −1 GDXH H −1 =X.
is defined as follows:
where I is the identity matrix,
Centrosymmetric matrices A and B satisfy the following conditions:
AB=BA.
is singular if the determinant of the matrix X, det(X)=0 (i.e., x4x1−x2x3=0). In this case, the inverse X−1 of the singular matrix X does not exist (division by zero). If the matrix X is singular, the matrix B=AX is also singular.
Singular Matrix Features:
Claims (6)
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US20210028929A1 (en) | 2021-01-28 |
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US20230254120A1 (en) | 2023-08-10 |
US20230254122A1 (en) | 2023-08-10 |
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